3 research outputs found
Lower bounds for the state complexity of probabilistic languages and the language of prime numbers
This paper studies the complexity of languages of finite words using automata
theory. To go beyond the class of regular languages, we consider infinite
automata and the notion of state complexity defined by Karp. Motivated by the
seminal paper of Rabin from 1963 introducing probabilistic automata, we study
the (deterministic) state complexity of probabilistic languages and prove that
probabilistic languages can have arbitrarily high deterministic state
complexity. We then look at alternating automata as introduced by Chandra,
Kozen and Stockmeyer: such machines run independent computations on the word
and gather their answers through boolean combinations. We devise a lower bound
technique relying on boundedly generated lattices of languages, and give two
applications of this technique. The first is a hierarchy theorem, stating that
there are languages of arbitrarily high polynomial alternating state
complexity, and the second is a linear lower bound on the alternating state
complexity of the prime numbers written in binary. This second result
strengthens a result of Hartmanis and Shank from 1968, which implies an
exponentially worse lower bound for the same model.Comment: Submitted to the Journal of Logic and Computation, Special Issue on
LFCS'2016) (Logical Foundations of Computer Science). Guest Editors: S.
Artemov and A. Nerode. This journal version extends two conference papers:
the first published in the proceedings of LFCS'2016 and the second in the
proceedings of LICS'2018. arXiv admin note: substantial text overlap with
arXiv:1607.0025
Irregular Behaviours for Probabilistic Automata
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